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ABOUT A STRAIN SMOOTHING TECHNIQUE IN FINITE ELEMENT METHOD Nguyen Xuan Hung 1 , Nguyen Dinh Hien 2 , Ngo Thanh Phong 1 1University of Natural Sciences, VNU – HCM Manuscript Received

ABOUT A STRAIN SMOOTHING TECHNIQUE

IN FINITE ELEMENT METHOD Nguyen Xuan Hung (1) , Nguyen Dinh Hien (2) , Ngo Thanh Phong (1)

(1)University of Natural Sciences, VNU – HCM

( Manuscript Received on April 15 th , 2007 )

ABSTRACT: This paper presents a global review of the strain smoothing method to finite element analysis for two-dimension elastostatics The strain at each point is replaced by a non – local approximation over a smoothing function With choosing a constant smoothed function and applying the divergence theorem, the stiffness matrix is calculated on boundaries of smoothing elements (smoothing cells) instead of their interior The presented method gains a high accuracy compared with the standard FEM without increasing computational cost

1.INTRODUCTION

In Finite Element Method (FEM), an important work to compute the stiffness matrix is

often to use mapped elements, such as the well-known isoparametric elements through Gauss

quadrature rule Then the element stiffness matrix is evaluated inside element instead of along boundaries of element In using a mapped element, a one – to – one coordinate transformation between the physical and natural coordinates of each element has to be ensured To satisfy this requirement, the convex element is not broken and a violently distorted mesh is not permitted Purpose of this paper is: 1) to construct the element stiffness matrix along its boundaries via a strain smoothing method, 2) to utilize a stabilized method with selective cell-wise strain smoothing when solving nearly incompressible elastic problems, 3) to estimate the reliability

of presented method through numerical examples

2 GOVERNING EQUATIONS AND WEAK FORM

acting within the domain The governing equilibrium equation for isotropic linear elasticity writes

1

2

T s

The displacement field satisfies the Dirichlet boundary conditions

i

i u

and the stress field satisfies the Newman boundary conditions

ij n j =t i

The virtual work equation is of the form

t

dev

1

e

n

e=

Ω ≈ Ω =UΩ The standard finite element solution uhof a finite element displacement model is expressed as follows

1

np h

i i i

N q

=

associated degrees of freedom at that node The discrete strain field is

s

operator)

By substituting Eq (8) - (9) into Eq (6), we obtain a linear system for q,

where the element stiffness matrix given by

μ

with

3

dev

−

1 1 0

1 1 0

0 0 0 λ

and the load vector is

3 THE STRAIN SMOOTHING METHOD

The strain smoothing method was proposed by Chen et al [1] and Yoo et al [3] as a

normalization of the local strain field This technique is also known as strain smoothing stabilization, through which the nodal strain is computed through the divergence of a spatial average of the standard local strain field In mesh-free methods [2], this is sufficient to eliminate defective modes through smoothed strains The derivatives of the shape functions are

not required at the nodes Applications of strain smoothing to the FEM can be seen as a stabilized conforming nodal integration method as defined in Galerkin mesh-free methods Strain smoothing at an arbitrary point writes

Ω

%

1/ ,

0,

e

C

A

⎪

∉Ω

⎪⎩

x

x x

x

(16)

Figure 1: Example of finite element meshes and smoothing cells

Substituting Eq (16) into Eq (15), and using the divergence theorem, we obtain

h h j

u

x

%

boundary

1

nb

b=

C

of edges of each smoothing cell The relationship between the strain field and nodal

The elemental stiffness matrix then writes

dev

C

matrix is in the form

1

C

T

C

x

C

shape functions are linear along the boundaries of the smoothing cells, one Gauss point is sufficient for exact integration of the weak form In this case,

1

1

b C

b

b

b

Considering a mixed variational principle based on an assumed strain field [4], the following system of linear algebraic equations is obtained

%

In strain smoothing technique, the element is subdivided into nc non-overlapping

sub-domains also called smoothing cells For example, the element is partitioned into 1-subcell, 2-subcell, 3-subcell and 4-subcell Then the strain is smoothed over each sub-cell While choosing a single subcell yields an element which is superconvergent in the H1 norm, and

insensitive to volumetric locking, as shown in Reference [6, 7], if nc > 1, locking reappears It

is also shown in Reference [6] that the finite element method with strain smoothing is

equivalent to a stress (equilibrium) formulated element for nc=1, and tends toward the

displacement results gradually improve, while stress results deteriorate

The purpose of this article is to use a single subcell smoothing to compute the volumetric part of the strain tensor, while the deviatoric strains are written in terms of an arbitrarily high

number of smoothing cells The method may be coined a stabilized method with selective

cell-wise strain smoothing [8]

The stiffness matrix is built

1 Using nc > 1 subcells to evaluate the deviatoric term

2 Using one single subcell to calculate the volumetric term

This leads to the following elemental stiffness matrix

1

nc

c dev c c c

=

whereA cis the area of the smoothing cell,ΩC

The resulting approach with selective smoothing cell brings out the stable and excellent convergence for compressible and nearly incompressible problems not only isotropic linear elasticity but also isotropic plasticity and viscoplasticity, etc

4.NUMERICAL RESULTS

4.1 Cantilever beam

A 2-D cantilever beam subjected to parabolic loading at the free end is examined in this

example as shown in Figure 3 The geometry is taken as length L, height D and thickness t,

solution of this problem is available as given by Reference [5] Figure 4 illustrates a uniform mesh with 512 quadrilateral elements

Figure 3 A cantilever beam and boundary conditions

The relative error in displacement norm is defined as

h exact exact

The error in energy is defined by

e

Ω

⎣∫ ε ε D ε ε ⎦

and the rate of convergence in the displacement norm for a sequence of uniform meshes, respectively

From Figures 4 – 5, the presented method gives reliable results compared with 4-node FEM Figures 4b and 5b show that the 2-Subcell, 3-Subcell and 4-Subcell elements exhibit the

displacement results for the 3-Subcell and 4-Subcell discretization are more accurate than the

P

L

D

y

x

A

standard bilinear Q4-FEM solution The proposed elements also produce a better approximation of the strain energy Additionally, the CPU time required for all elements presented here appears asymptotically lower than that of the FEM [6, 7], as the mesh size tends to zero

Figure 5 shows the convergence in energy and the rate of the cantilever beam Next we estimate the accuracy of the presented elements for the same beam problem, assuming a near incompressible material Under plane strain condition, Figure 6 illustrates the displacements along the neutral axis for Poisson’s ratio, ν = 0.4999 The results show that FEM, 2–subcell, 3–subcell and 4–subcell solutions yield poor accuracy as Poisson’s ratio ν tends toward 0.5 To remedy this locking phenomenon, selective integration techniques are considered Figure 6b presents the results after of the selective integration method (SIM) to Q4-FEM element and

using the selective cell-wise smoothing method for the FEM with strain smoothing [8]

Figure 4: The convergence in displacement norm; a) the relative error, b) the rate

Figure 5 The convergence of the energy norm; a) the energy, b) the rate

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Mesh index

FEM 1−Subcell 3−Subcell

0.0392

0.0394

0.0396

0.0398

0.04

0.0402

0.0404

Mesh index

Exact energy FEM 1−Subcell 3−Subcell

−2.5

−2

−1.5

−1

−0.5 0 0.5

FEM 1−Subcell 2−Subcell 3−Subcell

2.01

2.005

1.996

2.001 2.002

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

FEM 1−Subcell 3−Subcell

0.998

0.997

0.997 1.01

1.953

0 1 2 3 4 5 6 7 8

−3

−2.5

−2

−1.5

−1

−0.5

0x 10

−4

x (y=0)

Analytical solu SIM 1−Subcell 2−Subcell 4−Subcell

Figure 6: Vertical displacement for cantilever beam at the nodes along the x-axis (y =0): a) without

using the selective technique, b) applying the selective method (ν = 0.4999)

4.2 L-shaped domain with applied tractions

Consider a L – shaped domain under plane stress condition applied tractions and boundary

1

=

The convergence behaviour of overall strain energies is shown in Figure 8a, and the

The accuracy of presented method is higher than that of the FEM-Q4 The 1-Subcell element provides the best solutions in strain energy for the coarser meshes More particularly, an inversion of convergent energy for 2-Subcell and 3-Subcell is appeared Two these Sub-cells lead to the less error than 4-Subcell and FEM do Beside, a refined mesh towards to corner is

necessary for purposing the reduction of error and computational cost

0 1 2 3 4 5 6 7 8

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

−4

x (y=0)

Analytical solu.

FEM 1−Subcell 2−Subcell 3−Subcell 4−Subcell

Figure 7: Domain problem and initial mesh

0.5 1 1.5 2 2.5 3 3.5 4 4.5

1.51

1.52

1.53

1.54

1.55

1.56

1.57x 10

4

Mesh index

Estimated energy FEM 1−Subcell 3−Subcell

1 1.2 1.4 1.6 1.8 2 0.2

0.4 0.6 0.8 1 1.2 1.4 1.6

log10(Number of nodes) 1/2

FEM 1−Subcell 3−Subcell 4−Subcell

1.062 0.61

0.637 0.653 0.624

a) b)

Figure 8: The convergence of the energy norm; a) the strain energy, b) the rate

5 CONCLUSIONS

In this paper, we present a global aspect of the smoothed strains in finite element method

to solve compressible and incompressible linear elastic The element stiffness matrix is

completely calculated along the boundary instead of the inside of element that the traditional

FEM is utilized The shape functions are calculated in a simple form The numerical results

show that the present method is normally more accurate than FEM while computational cost is

not increasing

The method is illustrated in two – dimensional linear elasticity but it can be extended to

more complex structures such as non-linear elasticity, elastic – plastic behaviour and

viscoplastics, plates, shell, 3D-problems, etc The results of this investigation will be shown in

forthcoming papers

VỀ KỸ THUẬT TRƠN HÓA BIẾN DẠNG TRONG PHƯƠNG PHÁP PHẦN TỬ

HỮU HẠN Nguyễn Xuân Hùng (1) , Nguyễn Đình Hiển (2) , Ngô Thành Phong (1)

(1)Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM

(2)Trường Đại học Sư phạm Kỹ thuật Tp.HCM

TÓM TẮT: Bài báo này trình bày một tổng quan về phương pháp trơn hoá biến dạng

trong phần tử hữu hạn cho Cơ học vật rắn biến dạng hai chiều Biến dạng tại mỗi điểm được

chuẩn hóa bởi một hàm làm trơn trong một lân cận của miền khảo sát Khi hàm trơn được

chọn là hằng số, ma trận độ cứng được tính trên biên của phần tử thay vì bên trong như cách

tính thông thường Phương pháp đề cập đạt được độ chính xác cao hơn phương pháp phần tử

hữu hạn truyền thống mà không tăng chi phí tính toán

REFERENCES

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Galerkin mesh-free methods International Journal for Numerical Methods in

Engineering, 50:435 – 466 (2001)

[2] Belytschko T, Lu YY, Gu L Element – free Galerkin methods, International Journal

for Numerical Methods in Engineering, 37:229 – 256 (1994)

[3] Yoo JW, Moran B, Chen JS Stabilized conforming nodal integration in the

natural-element method International Journal for Numerical Methods in Engineering, 60:

861 – 890 (2004)

[4] Simo JC, Hughes TJR On the variational foundation of assumed strain methods

ASME Journal of Applied Mechanics, 53:51 – 54 (1986)

Hill, (1987)

[6] G.R Liu, K.Y Dai, T.T Nguyen, A smoothed finite element for mechanics problems,

Computational Mechanics, DOI 10.1007/s00466-006-0075-4, (2006)

[7] Nguyen X.H, Bordas S, and Nguyen-Dang H, Finite element methods with stabilized

conforming nodal integration: convergence, accuracy and properties, International

Journal Numerical Methods Engineering, submitted (2006)

[8] Nguyen X.H, Bordas S, Nguyen-Dang H Smooth strain finite elements: selective

integration, Collection of papers from Prof Nguyen-Dang Hung’s former students,

Vietnam National University –HCM Publishing House, 88 -106 (2006)